If the columns of A are linearly dependent, then det A = 0. TRUE (For example there is a row without a pivot so must be a row of all zeros.) det(A.start by defining the determinant via formulas that are nearly impossible to use except on We define the determinant det(A) of a square matrix as follows.Oct 7, 2013 If the matrix B = kA, i.e., every entries in A are multiplied by the constant k, then we iterated the previous theorem det(B) = kn det(A).So these two things are equivalent. So at least for the 2-by-2 case, the determinant of some matrix is equal to the determinant of the transpose of that matrix. Now .Two of the most important theorems about determinants are yet to be proved: Theorem 1: If A and B are both n × n matrices, then detAdetB = det(AB). Theorem.